The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools are meant to be used in different combinations on different problems and architectures. In this paper, we will describe these tools which include basic block matrix computations, the matrix sign function, 2-dimensional bisection, and spectral divide and conquer using the matrix sign function to find selected eigenvalues. We also outline how we deal with ill-conditioning and potential instability. Numerical examples are included. A future paper will discuss error analysis in detail and extensions to the generalized eigenproblem.

Abstract. A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate ofconvergence and it is shownhow reliable estimates ofthe optimal parametersmaybe computed in practice. To add to the known best approximation property of the unitary polar factor, the Hermitian polar factorHof a nonsingular Hermitian matrix A is shown to be a good positive definite approximation to A and 1/2(A/H) is shown to be a best Hermitian positive semi-definite approximation to A. Perturbation bounds for the polar factors are derived. Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix. Key words, polar decomposition, singular value decomposition, Newton’s method, matrix square root AMS(MOS) subject classifications. 65F25, 65F30, 65F35 1. Introduction. The

This paper catalogues formulas that are useful for estimating dynamic linear economic models. We describe algorithms for computing equilibria of an economic model and for recursively computing a Gaussian likelihood function and its gradient with respect to parameters. We apply these methods to several example economies.

Abstract. The standard inverse scaling and squaring algorithm for computing the matrix logarithm begins by transforming the matrix to Schur triangular form in order to facilitate subsequent matrix square root and Padé approximation computations. A transformation-free form of this method that exploits incomplete Denman–Beavers square root iterations and aims for a specified accuracy (ignoring roundoff) is presented. The error introduced by using approximate square roots is accounted for by a novel splitting lemma for logarithms of matrix products. The number of square root stages and the degree of the finalPadé approximation are chosen to minimize the computationalwork. This new method is attractive for high-performance computation since it uses only the basic building blocks of matrix multiplication, LU factorization and matrix inversion.

The sign function of a square matrix was introduced by Roberts in 1971. We show that it is useful to regard S = sign(A) as being part of a matrix sign decomposition A = SN , where N = (A 2 ) 1=2 . This decomposition leads to the new representation sign(A) = A(A 2 ) \Gamma1=2 . Most results for the matrix sign decomposition have a counterpart for the polar decomposition A = UH, and vice versa. To illustrate this, we derive best approximation properties of the factors U , H and S, determine bounds for kA \Gamma Sk and kA \Gamma Uk, and describe integral formulas for S and U . We also derive explicit expressions for the condition numbers of the factors S and N . An important equation expresses the sign of a block 2 \Theta 2 matrix involving A in terms of the polar factor U of A. We apply this equation to a family of iterations for computing S by Pandey, Kenney and Laub, to obtain a new family of iterations for computing U . The iterations have some attractive properties, includin...

Bjiirck and Hammarling [l] describe a fast, stable Schur method for computing a square root X of a matrix A (X2 = A). We present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix. For a nonsingular real matrix A conditions are given for the existence of a real square root, and for the existence of a real square root which is a polynomial in A; the number of square roots of the latter type is determined. The conditioning of matrix square roots is investigated, and an algorithm is given for the computation of a well-conditioned square root. 1.

. In this paper, preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues are presented. The basic mathematical theory behind this approach is reviewed and is followed by a discussion of the numerical considerations of the actual implementation. The numerical algorithm has been tested on thousands of matrices on both a Cray-2 and an IBM RS/6000 Model 580 workstation. The results of these tests are presented. Finally, issues concerning the parallel implementation of the algorithm are discussed. The algorithm's heavy reliance on matrix--matrix multiplication, coupled with the divide and conquer nature of this algorithm, should yield a highly parallelizable algorithm. Key words. eigenvalues, divide and conquer algorithm, invariant subspaces, parallel algorithm AMS subject classification. 65F15 PII. S1064827592228833 1. Introduction. Computation of all the eigenvalues and eigenvectors of a dens...

. A perturbation analysis shows that if a numerically stable procedure is used to compute the matrix sign function, then it is competitive with conventional methods for computing invariant subspaces. Stability analysis of the Newton iteration improves an earlier result of Byers and confirms that ill-conditioned iterates may cause numerical instability. Numerical examples demonstrate the theoretical results. 1. Introduction. If A 2 R n\Thetan has no eigenvalue on the imaginary axis, then the matrix sign function sign(A) may be defined as sign(A) = 1 ßi Z fl (zI \Gamma A) \Gamma1 dz \Gamma I; (1) where fl is any simple closed curve in the complex plane enclosing all eigenvalues of A with positive real part. The sign function is used to compute eigenvalues and invariant subspaces [2, 4, 6, 13, 14] and to solve Riccati and Sylvester equations [9, 15, 16, 28]. The matrix sign function is attractive for machine computation, because it can be efficiently evaluated by relatively simp...

this paper is the tradeoff between speed and stability. The single variable Newton iteration (1.2) is unstable, and the Pad e iteration (2.8) becomes unstable when we attempt to reduce the cost of its implementation. Iterations for the matrix square root appear to be particularly delicate with respect to numerical stability

Linear quadratic optimal control problems and the computation of Kalman filters require numerical solutions of discrete and continuous algebraic Riccati equations.